Finite Group Databasebeta

← #28599218

Subgroups — #28599218

Order 15104.

834 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837454C211yesmaximal, characteristic
2#21426492C211yesmaximal, characteristic
3#382×D59C211yesmaximal, characteristic
4#21424866C211yesmaximal, characteristic, complemented
5#21424864C211yesmaximal, characteristic, complemented
6#21424862C211yesmaximal, characteristic, complemented
7#21424290C211yesmaximal, characteristic
8#21424287C211yesmaximal, characteristic
9#21424285C211yesmaximal, characteristic
10#21424282C211yesmaximal, characteristic
11#21424259C211yesmaximal, characteristic, complemented
12#21424257C211yesmaximal, characteristic, complemented
13#21424225C211yesmaximal, characteristic, complemented
14#21424225C211yesmaximal, characteristic, complemented
15C59×#1949C211yesmaximal, characteristic, complemented
16C2×C4⋊C4×D59C2211yescharacteristic
17C2×C4⋊C4×D59C2211yescharacteristic
18C2×C42×D59C2211yescharacteristic
19#16837454C418yescomplemented
20C2×#14273661C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×(C2×D59)⋊C8C2211yescharacteristic
28C2×(C2×D59)⋊C8C2211yescharacteristic
29C2×C8×Dic59C2211yescharacteristic
30C2×C8×Dic59C2211yescharacteristic
31#16837269—21no—
32#16837268C2211yescharacteristic
33#16837267C2211yescharacteristic
34#16837267C2211yescharacteristic
35#16837267C2211yescharacteristic
36#16837267C2211yescharacteristic
37#16837266—21no—
38#16837266—21no—
39#16837264—21no—
40#16837259C2211yescharacteristic
41#16837259C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C4×(C2×D59)⋊C4—21no—
44#16837248—21no—
45#95×D59—21no—
46#95×D59—21no—
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×C4⋊C8×C59C2211yescharacteristic
56C59×#382C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×C4×D59—21no—
61C22×C4×D59—21no—
62C22×C4×D59D411yescharacteristic
63C22×C4×D59D411yescharacteristic
64C22×C4×D59C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C4⋊C4×D59—41no—
74C4⋊C4×D59—41no—
75C4⋊C4×D59—41no—
76C4⋊C4×D59—41no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59D411yescharacteristic
81C2×C4×Dic59D411yescharacteristic
82C2×C4×Dic59C2311yescharacteristic
83C42×D59C2×C414yes—
84C42×D59Q814yes—
85#14273661—41no—
86#14273660—41no—
87#14273660—41no—
88#14273659—41no—
89#14273658—41no—
90#14273658—41no—
91C4⋊(C59⋊C8)C2×C414yescomplemented
92(C2×D59)⋊C8C2×C414yes—
93(C2×D59)⋊C8C2×C414yes—
94C8×Dic59C2×C414yescomplemented
95C8×Dic59C2×C414yescomplemented
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C22×C8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C42×C59—21no—
116C2×C42×C59C2311yescharacteristic
117#107×C59—41no—
118#106×C59—41no—
119C4⋊C8×C59C2×C414yescomplemented
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123C23×D59C2×D411yescharacteristic
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2411yescharacteristic
131C22×Dic59—21no—
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59C2×D411yescharacteristic
136C22×Dic59C2×D411yescharacteristic
137C22×Dic59C2×D411yescharacteristic
138C2×C4×D59—22no—
139C2×C4×D59—22no—
140C2×C4×D59—21no—
141C2×C4×D59—21no—
142C2×C4×D59—21no—
143C2×C4×D59—21no—
144C2×C4×D59—41no—
145C2×C4×D59—41no—
146C2×C4×D59—42no—
147C2×C4×D59—42no—
148C2×C4×D59—42no—
149C2×C4×D59—42no—
150C2×C4×D59—42no—
151C2×C4×D59—42no—
152C2×C4×D59C4⋊C414yes—
153C2×C4×D59C4⋊C414yes—
154C2×C4×D59C22×C412yes—
155C2×C4×D59C2×Q812yes—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59C22×C412yes—
161C2×C8×C59C22×C412yes—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C42×C59—42no—
171C42×C59C22×C412yes—
172C42×C59C2×Q812yes—
173(C2×D59)⋊C4—42no—
174(C2×D59)⋊C4—42no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183Dic59⋊C4—42no—
184Dic59⋊C4—42no—
185C4×Dic59—42no—
186C4×Dic59C4⋊C414yes—
187C4×Dic59C4⋊C414yes—
188C4×Dic59C22×C412yes—
189C4×Dic59C2×Q812yes—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8C22×C412yes—
195C2×C59⋊C8C22×C412yes—
196C23×C59C22×D411yescharacteristic
197C22×D59—21no—
198C22×D59—21no—
199C22×D59—21no—
200C22×D59—21no—
201C22×D59C2×C4⋊C412yes—
202C22×D59#13512yes—
203C22×D59#13512yes—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—41no—
209C2×C4×C59—41no—
210C2×C4×C59—41no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59C2×C4⋊C412yes—
221C2×C4×C59C2×C4⋊C412yes—
222C2×C4×C59C23×C411yescharacteristic
223C2×C4×C59C22×Q811yescharacteristic
224C2×C4×C59H(5,𝔽2)11yescharacteristic
225C2×C4×C59H(5,𝔽2)11yescharacteristic
226C2×C4×C592−1+411yescharacteristic
227C2×C4×C592−1+411yescharacteristic
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59C2×C4⋊C412yes—
247C2×Dic59C2×C4⋊C412yes—
248C2×Dic59C2×C4⋊C412yes—
249C4×D59—28no—
250C4×D59—22no—
251C4×D59—22no—
252C4×D59—22no—
253C4×D59—22no—
254C4×D59—44no—
255C4×D59—84no—
256C4×D59—84no—
257C8×C59—22no—
258C8×C59—22no—
259C59⋊C8—22no—
260C59⋊C8—22no—
261#21983—591nomaximal, Sylow
262C22×C59C22×C4⋊C411yescharacteristic
263C22×C59#53511yescharacteristic
264C22×C59#53611yescharacteristic
265C22×C59C2×#13511yescharacteristic
266C22×C59C2×#13511yescharacteristic
267C22×C59#57511yescharacteristic
268C22×C59#57511yescharacteristic
269C2×D59—22no—
270C2×D59—22no—
271C2×D59—22no—
272C2×D59—22no—
273C2×D59—22no—
274C2×D59—22no—
275C2×D59#42714yes—
276C4×C59—22no—
277C4×C59—22no—
278C4×C59—81no—
279C4×C59—81no—
280C4×C59—81no—
281C4×C59—81no—
282C4×C59#52011yescharacteristic
283C4×C59#52011yescharacteristic
284C4×C59#54411yescharacteristic
285C4×C59#54411yescharacteristic
286Dic59—22no—
287Dic59—22no—
288Dic59—22no—
289Dic59—81no—
290Dic59—81no—
291Dic59—81no—
292Dic59—81no—
293C2×#431—591no—
294#2283—591no—
295C2×#382—591no—
296#1949—591no—
297#1949—591no—
298#1949—591no—
299#1949D5911yescharacteristic, complemented
300#1853—591no—
301#1853—591no—
302#1852—591no—
303#1852—591no—
304#1846—591no—
305#1846—591no—
306#1835—591no—
307#1835—591no—
308C2×C59#231711yescharacteristic
309C2×C59C2×#42711yescharacteristic
310C2×C59#291111yescharacteristic
311C2×C59#321111yescharacteristic
312C2×C59#321111yescharacteristic
313C2×C59#323211yescharacteristic
314C2×C59#323211yescharacteristic
315D59—24no—
316C22×C4⋊C4—591no—
317C22×C4⋊C4—591no—
318C22×C42—591no—
319#431—598no—
320C2×#107—591no—
321C2×#107—591no—
322C2×#107—591no—
323C2×#107C2×D5911yescharacteristic
324C2×#106—591no—
325C2×#106—591no—
326C2×#106—591no—
327C2×#106C2×D5911yescharacteristic
328C2×C4⋊C8—591no—
329C2×C4⋊C8C2×D5911yescharacteristic
330C2×C22⋊C8—591no—
331C2×C22⋊C8—591no—
332C2×C4×C8—591no—
333C2×C4×C8—591no—
334#383—1181no—
335#382—591no—
336#382—591no—
337#382—591no—
338#382—591no—
339#382—591no—
340#382—1181no—
341#382—1181no—
342#382C2×D5911yescharacteristic
343#381—1181no—
344#380—591no—
345#380—591no—
346C4×C4⋊C4—591no—
347C4×C4⋊C4C2×D5911yescharacteristic
348C4×C22⋊C4—1181no—
349#376—1181no—
350C2×#95—1181no—
351C2×#95—1181no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340C2×D5911yescharacteristic
359#340C2×D5911yescharacteristic
360C59#2198311yescharacteristic, complemented, Sylow
361C23×C4—591no—
362C23×C4—591no—
363C23×C4—591no—
364C23×C4—1181no—
365C23×C4—1181no—
366C22×C8—591no—
367C22×C8—591no—
368C22×C8C22×D5911yescharacteristic
369C22×C8C22×D5911yescharacteristic
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—2361no—
377C2×C4⋊C4—2361no—
378C2×C4⋊C4—2361no—
379C2×C4⋊C4—2361no—
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C4⋊C4C22×D5911yescharacteristic
382C2×C4⋊C4C22×D5911yescharacteristic
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C22⋊C4—1181no—
385C2×C22⋊C4—1181no—
386C2×C42—21no—
387C2×C42—591no—
388C2×C42—591no—
389C2×C42—591no—
390C2×C42—594no—
391C2×C42—594no—
392C2×C42—1181no—
393C2×C42C22×D5911yescharacteristic
394#107—41no—
395#107—2361no—
396#107—2361no—
397#107—2361no—
398#106—41no—
399#106—2361no—
400#106—2361no—
401#106—2361no—
402C4⋊C8—594no—
403C4⋊C8C4×D5914yescomplemented
404C22⋊C8—594no—
405C22⋊C8—594no—
406C4×C8—594no—
407C4×C8—594no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424C24—591no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—591no—
430C22×C4—591no—
431C22×C4—591no—
432C22×C4—594no—
433C22×C4—592no—
434C22×C4—592no—
435C22×C4—594no—
436C22×C4—1182no—
437C22×C4—1182no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—1181no—
442C22×C4—1181no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4—2361no—
447C22×C4—2361no—
448C22×C4—2362no—
449C22×C4—2362no—
450C22×C4—2362no—
451C22×C4—2362no—
452C22×C4—2362no—
453C22×C4—2362no—
454C22×C4D4×D5911yescharacteristic
455C22×C4D4×D5911yescharacteristic
456C22×C4C23×D5911yescharacteristic
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—592no—
462C2×C8—592no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8C2×C4×D5912yes—
468C2×C8C2×C4×D5912yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—2362no—
486C4⋊C4—2362no—
487C22⋊C4—2362no—
488C22⋊C4—2362no—
489C42—42no—
490C42—594no—
491C42—594no—
492C42—592no—
493C42—592no—
494C42—2362no—
495C42C2×C4×D5912yes—
496C42Q8×D5912yes—
497C23—592no—
498C23—592no—
499C23—592no—
500C23—1181no—
501C23—1181no—
502C23—1181no—
503C23—1181no—
504C23C2×D4×D5911yescharacteristic
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—592no—
522C2×C4—592no—
523C2×C4—592no—
524C2×C4—1188no—
525C2×C4—1181no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—1181no—
530C2×C4—1181no—
531C2×C4—1182no—
532C2×C4—1182no—
533C2×C4—1182no—
534C2×C4—1182no—
535C2×C4—4724no—
536C2×C4—4724no—
537C2×C4—2364no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4C4⋊C4×D5912yes—
551C2×C4C4⋊C4×D5912yes—
552C2×C4C22×C4×D5911yescharacteristic
553C2×C4C2×Q8×D5911yescharacteristic
554C2×C4#1427377211yescharacteristic
555C2×C4#1427377311yescharacteristic
556C2×C4#1427377411yescharacteristic
557C2×C4#1427377511yescharacteristic
558C8—22no—
559C8—22no—
560C8—1182no—
561C8—1182no—
562C22—594no—
563C22—1182no—
564C22—1182no—
565C22—1182no—
566C22—1182no—
567C22—1182no—
568C22—1182no—
569C22C2×C4⋊C4×D5911yescharacteristic
570C22#1683796811yescharacteristic
571C22#1683797511yescharacteristic
572C22#135×D5911yescharacteristic
573C22#135×D5911yescharacteristic
574C22#1683819811yescharacteristic
575C22#1683820111yescharacteristic
576C4—22no—
577C4—22no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—1182no—
583C4—1182no—
584C4—1182no—
585C4—4721no—
586C4—4721no—
587C4—4721no—
588C4—4721no—
589C4#1683788111yescharacteristic
590C4#1683788411yescharacteristic
591C4#1683802911yescharacteristic
592C4#1683803211yescharacteristic
593C2—1184no—
594C2#2142671811yescharacteristic
595C2#427×D5911yescharacteristic
596C2#2143268311yescharacteristic
597C2#2143505911yescharacteristic
598C2#2143506511yescharacteristic
599C2#2143526211yescharacteristic
600C2#2143526811yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1949C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1949D59
hypocenterC59#21983
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #21424866C2semidirect
5. #21424864C2semidirect
6. #21424862C2semidirect
11. #21424259C2semidirect
12. #21424257C2semidirect
13. #21424225C2semidirect
14. #21424225C2semidirect
15. C59×#1949C2semidirect
19. #16837454C4semidirect
91. C4⋊(C59⋊C8)C2×C4semidirect
94. C8×Dic59C2×C4semidirect
95. C8×Dic59C2×C4semidirect
119. C4⋊C8×C59C2×C4semidirect
299. #1949D59semidirect
360. C59#21983semidirect
403. C4⋊C8C4×D59semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168374542
#214264922
#382×D592
#214248662
#214248642
#214248622
#214242902
#214242872
#214242852
#214242822
#214242592
#214242572
#214242252
C59×#19492
#2198359